Affine Fractional Sobolev and Isoperimetric Inequalities

TL;DR

This paper establishes new sharp affine fractional Sobolev inequalities that improve upon previous inequalities, connect to affine isoperimetric inequalities, and prove a conjecture related to radial mean bodies.

Contribution

The paper introduces stronger affine fractional Sobolev inequalities for all 0<s<1, linking them to existing inequalities and proving a conjecture about radial mean bodies.

Findings

New sharp affine fractional Sobolev inequalities for 0<s<1.

Fractional Petty projection inequalities stronger than Euclidean isoperimetric inequalities.

Proof of a conjecture for radial mean bodies.

Abstract

Sharp affine fractional Sobolev inequalities for functions on $\mathbb R^n$ are established. For each $0<s<1$, the new inequalities are significantly stronger than (and directly imply) the sharp fractional Sobolev inequalities of Almgren and Lieb. In the limit as $s\to 1^-$, the new inequalities imply the sharp affine Sobolev inequality of Gaoyong Zhang. As a consequence, fractional Petty projection inequalities are obtained that are stronger than the fractional Euclidean isoperimetric inequalities and a natural conjecture for radial mean bodies is proved.